Okay! So I know the answer to this one, because I AM Chinese (Mandarin speaking), attended Chinese school, my mother still teaches Chinese school, and my uncle is actually principal of a major Chinese school in San Gabriel Valley.
The answer to your questions is yes, even a Cantonese speaker would take Mandarin Chinese classes (and, btw, even in Hong Kong now everyone has to learn Mandarin. And even growing up, not that I've met every Cantonese speaker ever, but I've never met a Cantonese speaker in America who can't at least get by in Mandarin).
School does take place on Saturdays, usually for a half day, generally at local high schools. The teachers are dedicated, but sometimes not very good; they don't always have education backgrounds, for example. I've never really met anyone who was fluent just through Chinese school, because honestly, 4 hours a week isn't enough to get fluent in the language, especially learning to read and write it (most of us ABCs can speak it fine, because we speak it at home, but Chinese isn't phonetic so it's hard to learn to read/write).
For the most part in my community, kids start dropping out as they enter high school because of lack of time (and also, because nowadays many high schools offer daily Chinese classes + AP Chinese).
Oh yeah, and Chinese school is boring as [Satan's home]
If you have any other questions about being an ABC or Chinese school in general, ask away or PM me!